System and method for measuring depth and velocity of instrumentation within a wellbore using a bendable tool

ABSTRACT

An apparatus and method for measuring depth, velocity, or both depth and velocity of instrumentation along a wellbore is provided. The apparatus includes a downhole portion movable within the wellbore in a direction generally parallel to the wellbore. The apparatus further includes a first acceleration sensor that generates a first signal indicative of a first acceleration. The apparatus further includes a second acceleration sensor that generates a second signal indicative of a second acceleration. The apparatus further includes a bend sensor generating a third signal indicative of an amount of bend of at least a portion of the downhole portion.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.13/245,435, filed on Sep. 26, 2011 which is incorporated in its entiretyby reference herein, which is a continuation of U.S. patent applicationSer. No. 11/866,213, filed Oct. 2, 2007, which is incorporated in itsentirety by reference herein.

BACKGROUND

1. Field of the Invention

The present application relates generally to systems and methods fordetermining the depth, the velocity, or both the depth and the velocityof an instrumentation package within a wellbore.

2. Description of the Related Art

Surface-based wellbore depth measurements are typically madeperiodically during wellbore drilling for the exploration of oil and gasdeposits to determine the absolute depth of the drilling tool within thewellbore. In such measurements, the depth of the drilling tool istypically determined by surface measurements of the lengths of the pipesections inserted into the wellbore between the drilling tool and thesurface.

Using wireline surveys, the drilling of the wellbore is periodicallyhalted and a survey tool is lowered into the wellbore. As the surveytool is guided along the wellbore, it can provide information regardingits orientation and location by sending signals through a wire or cableto the surface. The absolute depth of the survey tool down the wellboreis typically given by a surface-based measurement of the length of thewire or cable between the survey tool and the surface. Similarly,surface-based logging measurements of the absolute depth of detectedgeological formations are typically made by surface measurements of thelength of the wire or cable between the logging tool and the surface.However, due to various distortions, the cumulative length of the pipesections or of the wire while within the wellbore can differ from thecumulative length measured at the surface, resulting in errors in thedetermination of the absolute depth.

In addition, surface-based measurements of the velocity of the surveytool along the wellbore can be used to determine the relative distancesbetween detected features or formations within the wellbore. Forexample, the time period between detecting two separate features alongthe wellbore and the velocity of the survey tool during this time periodcan be multiplied together to provide the relative distance between thetwo detected features. However, as with the surface-based depthmeasurements described above, surface-based velocity measurements do notprovide a sufficient accuracy (e.g., within only a few centimeters) totell when two geophysical sensors with significant along-hole separationpass the same geological formation within only a few centimeters ofaccuracy. For example, due to friction or other effects within thewellbore, the survey tool can move in a jerking manner with varyingvelocity. In addition, these effects can result in the velocity of thesurvey tool within the wellbore differing from the measured velocity ofthe wire or cable at the surface.

Inertial navigation systems have been proposed as being able to provideimproved wellbore depth and velocity measurements. Such inertialnavigation systems can determine the depth of the survey tool by doubleintegration of the detected acceleration. However, such procedures arevulnerable to errors in the detected acceleration, which results in adrift of the depth measurement from the true depth of the survey tool.It has been difficult, and will likely continue to be difficult, to getsuch inertial navigation systems to work without providing aiding datasuch as surface-based depth measurements or updates of the depthobtained while the survey tool is stationary (i.e., zero-velocityupdates) to remove this drift. However, it is generally desirable toavoid using surface-based aiding data, since such data would effectivelytransform an inertial navigation system into a surface-depth system,thereby including the problems of such systems. It is also generallydesirable to avoid zero-velocity updates due to large increases in thetime and costs of conducting such surveys.

Improved wellbore depth and velocity measurements are desirable tounderstand the geological formations being drilled and the oil or gasdeposits being accessed. For example, improved measurements can resolveuncertainties in the depth measurements of a geological fault fromwellbore positioning surveys in two nearby wells. In addition, improvedwellbore depth measurements are helpful for drilling safety by providingmore reliable information regarding the true wellbore depth to avoiddrilling into adjacent wells.

SUMMARY

In certain embodiments, an apparatus for use in a wellbore is provided.The apparatus comprises a downhole portion movable within the wellborein a direction generally parallel to the wellbore. The apparatus furthercomprises a first acceleration sensor mounted at a first position withinthe downhole portion. The first acceleration sensor generates a firstsignal indicative of a first acceleration in a first direction generallyparallel to the wellbore at the first position. The apparatus furthercomprises a second acceleration sensor mounted at a second positionwithin the downhole portion. The second acceleration sensor generates asecond signal indicative of a second acceleration in a second directiongenerally parallel to the wellbore at the second position. The apparatusfurther comprises a bend sensor generating a third signal indicative ofan amount of bend of at least a portion of the downhole portion.

In certain embodiments, a method generates information indicative of adepth or a velocity, or both a depth and a velocity, of a downholeportion of a tool movable within a wellbore. The method comprisesproviding a tool comprising a downhole portion, a first accelerationsensor, a second acceleration sensor, and a bend sensor. The downholeportion is movable within the wellbore in a direction generally parallelto the wellbore. The first acceleration sensor is mounted at a firstposition within the downhole portion. The first acceleration sensorgenerates a first signal indicative of a first acceleration in a firstdirection generally parallel to the wellbore at the first position. Thesecond acceleration sensor is mounted at a second position within thedownhole portion. The second acceleration sensor generates a secondsignal indicative of a second acceleration in a second directiongenerally parallel to the wellbore at the second position. The bendsensor generates a third signal indicative of an amount of bend of atleast a portion of the downhole portion. The method further comprisesgenerating the first signal, the second signal, and the third signalwhile the downhole portion is at a first location within the wellbore.The method further comprises generating the first signal, the secondsignal, and the third signal while the downhole portion is at a secondlocation within the wellbore.

In certain embodiments, a method determines a depth or a velocity, orboth a depth and a velocity, of a downhole portion of a tool movablewithin a wellbore. The method comprises receiving one or moreacceleration measurements from at least one acceleration sensor in thedownhole portion of the tool. The method further comprises receiving oneor more measurements of an amount of bend of at least a portion of thedownhole portion. The method further comprises calculating a depth, or avelocity, or both a depth and a velocity, of the downhole portion of thetool in response to the one or more acceleration measurements and theone or more measurements of the amount of bend.

In certain embodiments, a bendable tool for use in a wellbore isprovided. The tool comprises a first acceleration sensor mounted withinthe tool. The first acceleration sensor is configured to generate afirst signal indicative of a first acceleration of the firstacceleration sensor in a first direction generally parallel to thewellbore. The tool further comprises a second acceleration sensormounted within the tool. The second acceleration sensor is configured togenerate a second signal indicative of a second acceleration of thesecond acceleration sensor in a second direction generally parallel tothe wellbore. The tool further comprises a bend sensor configured togenerate a third signal indicative of an amount of bend of at least aportion of the tool between the first acceleration sensor and the secondacceleration sensor.

In certain embodiments, a system for use with an apparatus configured tobe inserted into a wellbore is provided. The system comprises one ormore inputs configured to receive a first signal from a firstacceleration sensor of the apparatus, a second signal from a secondacceleration sensor of the apparatus, and a third signal from a bendsensor of the apparatus. The first acceleration sensor is within adownhole portion of the apparatus, and the second acceleration sensor iswithin the downhole portion of the apparatus and spaced from the firstacceleration sensor generally along the wellbore. The first signal isindicative of a first acceleration of the first acceleration sensor in afirst direction generally parallel to the wellbore. The second signal isindicative of a second acceleration of the second acceleration sensor ina second direction generally parallel to the wellbore. The third signalis indicative of an amount of bend of at least a portion of theapparatus between the first acceleration sensor and the secondacceleration sensor. The tool further comprises a controller configuredto calculate a depth, a velocity, or both a depth and a velocity of thedownhole portion within the wellbore in response to the first signal,the second signal, and the third signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates an example survey tool compatible withcertain embodiments described herein for use in a wellbore.

FIG. 2 schematically illustrates an example survey tool in a portion ofthe wellbore having a curvature.

FIG. 3 schematically illustrates an example survey tool as part of alogging assembly having a first supplementary sensor and a secondsupplementary sensor.

FIG. 4A is a flowchart of an example method of determining a depth of adownhole portion of a survey tool in accordance with certain embodimentsdescribed herein.

FIG. 4B is a flowchart of an example method of determining a velocity ofa downhole portion of a survey tool in accordance with certainembodiments described herein.

FIG. 5A schematically illustrates an example survey tool having a firstsupplementary sensor and a second supplementary sensor passing alandmark feature.

FIG. 5B is an example plot of the signals from the first supplementarysensor and the second supplementary sensor as a function of time.

FIG. 6 schematically illustrates an example survey tool having a bendsensor between the first supplementary sensor and the secondsupplementary sensor in accordance with certain embodiments describedherein.

FIGS. 7A and 7B schematically illustrate an example bend sensorutilizing a laser beam and a light sensitive target in an unbent end abent configuration, respectively, in accordance with certain embodimentsdescribed herein.

FIG. 8 is a flowchart of an example method of determining a depth of adownhole portion of a survey tool having a bend sensor in accordancewith certain embodiments described herein.

FIG. 9 is a flowchart of an example method of determining a velocity ofa downhole portion of a survey tool having a bend sensor in accordancewith certain embodiments described herein.

DETAILED DESCRIPTION

Certain embodiments described herein provide a true downhole-basedsystem for measuring a depth, a velocity, or both a depth and a velocityof a downhole portion with sufficient accuracy for logging and drillingapplications.

FIG. 1 schematically illustrates an example survey tool 10 compatiblewith certain embodiments described herein for use in a wellbore 20. Thesurvey tool 10 comprises a downhole portion 30 having an axis 32. Thedownhole portion 30 is adapted to move within the wellbore 20 with theaxis 32 generally parallel to the wellbore 20. The survey tool 10further comprises a first acceleration sensor 40 mounted at a firstposition 42 within the downhole portion 30. The first accelerationsensor 40 is adapted to generate a first signal indicative of anacceleration of the first acceleration sensor 40 along the axis 32. Thesurvey tool 10 further comprises a second acceleration sensor 50 mountedat a second position 52 within the downhole portion 30. The secondposition 52 is spaced from the first position 42 by a non-zero distanceB along the axis 32. The second acceleration sensor 50 is adapted togenerate a second signal indicative of an acceleration of the secondacceleration sensor 50 along the axis 32. The survey tool 10 furthercomprises a controller 60 adapted to receive the first signal and thesecond signal. The controller 60 is further adapted to calculate adepth, a velocity, or both a depth and a velocity of the downholeportion 30 in response to the first signal and the second signal. In theembodiment schematically illustrated by FIG. 1, the controller 60 is atthe surface and is coupled to the downhole portion 30 by a cable 62.

In certain embodiments, the survey tool 10 is a component of a drillstring and is used to determine the actual depth of a drilling tool(e.g., drill bit) of the drilling assembly. Drill strings compatiblewith embodiments described herein include, but are not limited to,measurement-while-drilling (MWD) strings. In certain other embodiments,the survey tool 10 is a component of a navigational string and is usedto determine at least a portion of the wellbore path. In certain otherembodiments, the survey tool 10 is a component of a logging string andis used to determine the actual depth of detected geological featuresalong the wellbore 20 or relative depths between detected geologicalfeatures along the wellbore 20. Logging strings compatible withembodiments described herein include, but are not limited tologging-while-drilling (LWD) strings. In certain embodiments, the drillstring or the logging string includes a sufficient number of sensors andadequate spacings between the first acceleration sensor 40 and thesecond acceleration sensor 50 to perform the method described below. Incertain embodiments, the drill string or the logging string includesadditional acceleration sensors (e.g., cross-axial accelerometers) thatcan be used to provide measurements for the determination of theinclination and high-side toolface angle of the downhole instrumentationat intervals along the well path trajectory.

In certain embodiments, the downhole portion 30 comprises a housing 64containing at least one of the acceleration sensors. As schematicallyillustrated by FIG. 1, the housing 64 of certain embodiments containsboth the first acceleration sensor 40 and the second acceleration sensor50. In other embodiments, the first acceleration sensor 40 and thesecond acceleration sensor 50 are not contained in a single housing, butare positioned on different portions of the downhole portion 30. Incertain embodiments, the downhole portion 30 further comprises portions(not shown), such as collars or extensions, which contact an innersurface of the wellbore 20 to position the housing 64 substantiallycollinearly with the wellbore 20.

In certain embodiments, the downhole portion 30 is adapted to bend asthe downhole portion 30 moves through a curved portion of the wellbore20. FIG. 2 schematically illustrates an example survey tool 10 having adownhole portion 30 within a section of the wellbore 20 having acurvature such that the direction of the wellbore 20 changes by anon-zero angle θ. In certain such embodiments, the downhole portion 30bends by the non-zero angle θ such that the axis 32 of the downholeportion 30 is substantially parallel to the wellbore 20. Under suchconditions, the axis 32 at the first position 42 is at the non-zeroangle with respect to the axis 32 at the second position 52.

In certain embodiments, the first acceleration sensor 40 and the secondacceleration sensor 50 comprise accelerometers currently used inconventional wellbore survey tools. In certain embodiments, one or bothof the first acceleration sensor 40 and the second acceleration sensor50 comprise a single-axis accelerometer sensitive to accelerations alonga single sensing direction. In certain such embodiments, the single-axisaccelerometer is advantageously mounted so that its sensing direction issubstantially parallel with the axis 32 of the downhole portion 30. Inother embodiments, one or both of the first acceleration sensor 40 andthe second acceleration sensor 50 comprise a two-axis or a three-axisaccelerometer sensitive to accelerations in multiple directions (e.g., amultiple-axis accelerometer). For example, a three-axis accelerationsensor can be used capable of measuring accelerations along the axis ofthe downhole portion 30 and in two generally orthogonal directions in aplane (e.g., a cross-axial plane) that is generally perpendicular to theaxis of the downhole portion 30. In certain embodiments, the x and yaxes are defined to lie in the cross-axial plane while the z axis iscoincident with the axis of the wellbore 20 or the downhole portion 30.In certain such embodiments, the multiple-axis accelerometer isadvantageously mounted so that it is sensitive to accelerations along atleast one direction parallel to the axis 32 of the downhole portion 30.In certain embodiments, the first acceleration sensor 40 and the secondacceleration sensor 50 are advantageously substantially identical.Example accelerometers include, but are not limited to, quartz flexuresuspension accelerometers available from a variety of vendors. Othertypes of acceleration sensors are also compatible with certainembodiments described herein.

In certain embodiments, the distance B between the first position 42 andthe second position 52 along the axis 32 of the downhole portion 30 isadvantageously selected to be long enough to respond to curvature of thewellbore 20 such that the downhole portion 30 bends substantiallyequally to the curvature of the wellbore 20. The distance B of certainembodiments is selected such that the first acceleration sensor 40 andthe second acceleration sensor 50 detect different accelerations alongthe axis 32 when the downhole portion 30 is in a curved portion of thewellbore 20. In certain embodiments, the distance B is larger thanapproximately 10 meters, while in other embodiments, the distance B isin a range between approximately 10 meters and approximately 30 meters.Other distances B are compatible with certain embodiments describedherein.

In certain embodiments, the downhole portion 30 comprises one or moresupplementary sensors in addition to the first acceleration sensor 40and the second acceleration sensor 50. FIG. 3 schematically illustratesan example survey tool 10 comprising a first supplementary sensor 70 anda second supplementary sensor 80. Example supplementary sensors of thedownhole portion 30 include, but are not limited to, gamma-ray sensorsadapted to detect gamma rays from geological formations in proximity tothe downhole portion 30 and magnetic sensors adapted to detect casingcollars of the pipe casing sections of the wellbore 20 in proximity tothe downhole portion 30. In the embodiment schematically illustrated byFIG. 3, the first supplementary sensor 70 is below and in proximity tothe first acceleration sensor 40 and the second supplementary sensor 80is above and in proximity to the second acceleration sensor 50. Otherembodiments can have other configurations of at least one supplementarysensor and the two acceleration sensors. In certain embodiments, suchsupplementary sensors are used in conjunction with the survey tool 10,as described more fully below, to provide additional data which is usedto aid the determination of the depth, the velocity, or both the depthand the velocity of the downhole portion 30 within the wellbore 20.

In certain embodiments, the controller 60 is adapted to determine thedepth, the velocity, or both the depth and the velocity of the downholeportion 30 in response to signals received from the various sensors ofthe downhole portion 30. In certain embodiments, the controller 60comprises a microprocessor adapted to perform the method described belowfor determining the depth, the velocity, or both the depth and thevelocity of the downhole portion 30 within the wellbore 20. In certainembodiments, the controller 60 further comprises a memory subsystemadapted to store at least a portion of the data obtained from thevarious sensors. The controller 60 can comprise hardware, software, or acombination of both hardware and software to accomplish the calculationof the depth, the velocity, or both the depth and the velocity. Incertain embodiments, the controller 60 comprises a standard personalcomputer.

In certain embodiments, at least a portion of the controller 60 islocated within the downhole portion 30. In certain other embodiments, atleast a portion of the controller 60 is located at the surface and iscoupled to the downhole portion 30 within the wellbore 20 by a wire orcable 62. In certain such embodiments, the cable 62 comprises signalconduits through which signals are transmitted from the various sensorswithin the downhole portion 30 to the controller 60. In certainembodiments in which the controller 60 is adapted to generate controlsignals for the various components of the survey tool 10 in the downholeportion 30, the cable 62 is adapted to transmit the control signals fromthe controller 60 to the downhole portion 30. In certain embodiments inwhich the downhole portion 30 is part of a wellbore drilling systemcapable of measurement while drilling (MWD) or logging while drilling(LWD), signals from the downhole portion 30 are transmitted by mud pulsetelemetry using rapid fluctuations in the pressure of a closed loopcirculating system or electromagnetic (EM) telemetry.

In certain embodiments, the controller 60 is adapted to perform apost-processing analysis of the data obtained from the various sensorsof the survey tool 10. In certain such post-processing embodiments, datais obtained and saved from the various sensors of the survey tool 10 asthe downhole portion 30 travels within the wellbore 20, and the saveddata are later analyzed to determine relative depths and/or absolutedepths of the various detected features. The saved data obtained fromthe various sensors, including any aiding data (described more fullybelow) advantageously includes time reference information (e.g., timetagging) so that the relative times of the detection of various featurescan be determined. In certain embodiments, the relevant data from thevarious sensors are manually inspected and correlated with one anotherto provide aiding data. In other embodiments, the controller 60 performsthis correlation of the saved data automatically to provide the aidingdata.

In certain other embodiments, the controller 60 provides a real-timeprocessing analysis of the signals or data obtained from the varioussensors of the survey tool 10. In certain such real-time processingembodiments, data obtained from the various sensors of the survey tool10 are analyzed while the downhole portion 30 travels within thewellbore 20. In certain embodiments, at least a portion of the dataobtained from the various sensors is saved in memory for analysis by thecontroller 60. The controller 60 of certain such embodiments comprisessufficient data processing and data storage capacity to perform thereal-time analysis. In certain embodiments, the relevant data from thevarious sensors are advantageously correlated with one another by thecontroller 60 to provide aiding data. In certain embodiments, theprocessing analysis can include a recursive estimation algorithm.

FIG. 4A is a flowchart of an example method 100 for determining a depthof a downhole portion 30 of a survey tool 10 along a wellbore 20 inaccordance with certain embodiments described herein. While the method100 is described herein by reference to the survey tool 10 schematicallyillustrated by FIGS. 1-3, other survey tools 10 are also compatible withembodiments of the method 100.

In certain embodiments, the method 100 comprises providing the surveytool 10 comprising a downhole portion 30 within the wellbore 20 in anoperational block 110. The downhole portion 30 comprises a firstacceleration sensor 40 and a second acceleration sensor 50. The firstacceleration sensor 40 is adapted to generate a first signal indicativeof an acceleration of the first acceleration sensor 40 along thewellbore 20. The second acceleration sensor 50 is adapted to generate asecond signal indicative of an acceleration of the second accelerationsensor 50 along the wellbore 20. The second acceleration sensor 50 isspaced from the first acceleration sensor 40 by a non-zero distance.

In certain embodiments, the method 100 further comprises receiving thefirst signal and the second signal while the downhole portion 30 is at afirst location within the wellbore 20 in an operational block 120. Thedownhole portion 30 of certain embodiments is stationary while at thefirst location, while in other embodiments, the downhole portion 30 ismoving along the wellbore 20 while at the first location.

In certain embodiments, the method 100 further comprises receiving thefirst signal and the second signal while the downhole portion 30 is at asecond location within the wellbore 20 in an operational block 130. Thedownhole portion 30 of certain embodiments is stationary while at thesecond location, while in other embodiments, the downhole portion 30 ismoving along the wellbore 20 while at the second location.

In certain embodiments, the method 100 further comprises calculating adepth of the downhole portion 30 in an operational block 140. The depthis calculated in response to the first signal and the second signalreceived while the downhole portion 30 is at the first location and inresponse to the first signal and the second signal received while thedownhole portion 30 is at the second location.

FIG. 4B is a flowchart of an example method 200 for determining avelocity of a downhole portion 30 of a survey tool 10 between twolocations along a wellbore 20 in accordance with certain embodimentsdescribed herein. While the method 200 is described herein by referenceto the survey tool 10 schematically illustrated by FIGS. 1-3, othersurvey tools 10 are also compatible with embodiments of the method 200.

In certain embodiments, the method 200 comprises providing the surveytool 10 comprising a downhole portion 30 in an operational block 210.The downhole portion 30 comprises a first acceleration sensor 40 and asecond acceleration sensor 50. The first acceleration sensor 40 isadapted to generate a first signal indicative of an acceleration of thefirst acceleration sensor 40 along the wellbore. The second accelerationsensor 50 is adapted to generate a second signal indicative of anacceleration of the second acceleration sensor 50 along the wellbore.The second acceleration sensor 50 is spaced from the first accelerationsensor 40 by a non-zero distance.

In certain embodiments, the method 200 further comprises receiving thefirst signal and the second signal while the downhole portion 30 is at afirst location within the wellbore 20 in an operational block 220. Thedownhole portion 30 of certain embodiments is moving along the wellbore20 while at the first location. In certain embodiments, the method 200further comprises receiving the first signal and the second signal whilethe downhole portion 30 is at a second location within the wellbore 20in an operational block 230. The downhole portion 30 of certainembodiments is moving along the wellbore 20 while at the secondlocation.

In certain embodiments, the method 200 further comprises calculating avelocity of the downhole portion 30 between the first location and thesecond location in an operational block 240. The velocity is calculatedin response to the first signal and the second signal received while thedownhole portion 30 is at the first location and in response to thefirst signal and the second signal received while the downhole portion30 is at the second location.

An example embodiment for determining the depth, the velocity, or boththe depth and the velocity of a downhole portion 30 of a survey tool 10utilizing a first acceleration sensor 40 and a second accelerationsensor 50 is described below. While the example embodiment describedbelow has a minimum number of variables, other embodiments are notlimited to only these variables. Additional variables may also be used,including, but not limited to, misalignments of the acceleration sensorsrelative to the axis 32. In certain embodiments, the units of theparameters and variables below are in meters-kilogram-second (MKS)units.

Aiding data from other downhole supplementary sensors (e.g., gamma-raysensors, magnetic sensors for locating casing collars) is advantageouslyincluded in certain embodiments to enhance the resultant accuracy of theresults. Other embodiments do not utilize aiding data or suchsupplementary sensors. The example embodiment described below includesthe use of aiding data, such as velocity data, absolute-depth data, andrelative-depth data. Other types and/or combinations of aiding data arealso used in certain other embodiments.

In the example embodiment described below, the periodicity of themeasurements from the two accelerometers define time periods or “epochs”whereby one set of accelerometer measurements are taken at every epochk. Aiding data are taken in the example embodiments only at a subset ofthese epochs. In certain embodiments, the different types of aiding dataare taken the same epochs, while in other embodiments, the differenttypes of aiding data are taken at different epochs.

In the example embodiment described below, the first acceleration sensor40 is referred to as the “upper acceleration sensor” and the secondacceleration sensor 50 is referred to as the “lower accelerationsensor.” The terms “upper” and “lower” are used herein merely todistinguish the two acceleration sensors according to their relativepositions along the wellbore 20, and are not to be interpreted aslimiting. Other embodiments distinguish the two acceleration sensorsfrom one another using other terms.

Example Embodiment State Vector with Five Elements

The example embodiment described below utilizes a state vector havingfive elements and the following parameters:

-   -   g=magnitude of gravity;    -   Δt=time between updates; and    -   B=distance between the upper acceleration sensor 40 (denoted        below by the subscript “U”) and the lower acceleration sensor 50        (denoted below by the subscript “L”).        In certain embodiments, the time between updates Δt is        synchronized with the clock frequency of the computer system        used to perform the example embodiment.

The state vector X_(k) at epoch k is expressed as follows:

X _(k) =[a _(k) v _(k) D _(L,k) d _(k) I _(L,k)]^(T);  (Eq. 1)

where a_(k) is the calculated acceleration of the survey tool 10 in adirection generally parallel to the wellbore 20, v_(k) is the calculatedvelocity of the survey tool 10 in a direction generally parallel to thewellbore 20, D_(L,k) is the calculated depth of the lower accelerationsensor 50, d_(k) is the calculated apparent dogleg, which is equal tothe difference between the inclinations of the lower acceleration sensor50 and the upper acceleration sensor 40, divided by the distance B(i.e., d_(k)=(I_(L,k)−I_(U,k))/B), and I_(L,k) is the calculatedinclination of the lower acceleration sensor 50, assuming that azimuthchanges between the upper and lower acceleration sensors are relativelysmall, which is true in certain embodiments described herein.

The state co-variance matrix at epoch k is expressed as follows:

$\begin{matrix}{{\Sigma_{k} = \begin{bmatrix}\sigma_{a,k}^{2} & \sigma_{{av},k} & \sigma_{{aD},k} & \sigma_{{ad},k} & \sigma_{{aI},k} \\\sigma_{{va},k} & \sigma_{v,k}^{2} & \sigma_{{vD},k} & \sigma_{{vd},k} & \sigma_{{vI},k} \\\sigma_{{Da},k} & \sigma_{{Dv},k} & \sigma_{D,k}^{2} & \sigma_{{Dd},k} & \sigma_{{DI},k} \\\sigma_{{da},k} & \sigma_{{dv},k} & \sigma_{{dD},k} & \sigma_{d,k}^{2} & \sigma_{{dI},k} \\\sigma_{{Ia},k} & \sigma_{{Iv},k} & \sigma_{{ID},k} & \sigma_{{Id},k} & \sigma_{I,k}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 2} )\end{matrix}$

where σ² _(i,k) is the variance of parameter number i in state vectorX_(k), and is the co-variance between parameter number i and j in statevector X_(k).

The initial state at epoch k=0, corresponding to the survey tool 10 in astationary condition within the wellbore 20 is given by the following:

X ₀=[0 0 D _(L,0)(I _(L,0) −I _(U,0))/B I _(L,0)]^(T);  (Eq. 3)

where D_(L,0) is the initial depth of the lower acceleration sensor 50,I_(L,0) is the initial inclination of the lower acceleration sensor 50,and I_(U,0) is the initial inclination of the upper acceleration sensor40.

In certain embodiments, the initial depth is referred to a known pointin the wellbore 20. The initial inclinations of certain embodiments isdetermined from stationary acceleration sensor measurements or fromknown wellbore geometry of a landmark location. Additional accelerationsensors are compatible with the use of acceleration-sensor-based initialinclinations. In certain embodiments, a pair of high-side cross-axialacceleration sensors can be used to provide initial stationaryacceleration measurements in a direction substantially perpendicular tothe axis 32 and having a component in a vertical plane. In otherembodiments, the first acceleration sensor 40 and the secondacceleration sensor 50 each comprises a multiple-axis accelerometer(e.g., a two-axis or a three-axis accelerometer) that provides signalsindicative of the acceleration parallel to the axis 32 and theacceleration in a direction substantially perpendicular to the axis 32and having a component in a vertical plane. In certain such embodiments,the initial inclinations are then given by:

I _(L,0)=arctan(−HA _(L,0) /A _(L,0));  (Eq. 4)

I _(U,0)=arctan(−HA _(U,0) /A _(U,0));  (Eq. 5)

where A_(L,0) is the initial lower acceleration sensor measurement in adirection generally parallel to the axis 32, A_(U,0) is the initialupper acceleration sensor measurement in a direction generally parallelto the axis 32, HA_(L,0) is the initial lower acceleration sensormeasurement in a direction substantially perpendicular to the axis 32and having a component in a vertical plane (i.e., the lower high-sideacceleration), and HA_(U,0) is the initial upper acceleration sensormeasurement in a direction substantially perpendicular to the axis 32and having a component in a vertical plane (i.e., the upper high-sideacceleration).

The co-variance matrix L for the initial state at epoch k=0 can beexpressed as the following:

$\begin{matrix}{{\Sigma_{0} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \sigma_{D}^{2} & 0 & 0 \\0 & 0 & 0 & {\sigma_{I}^{2}/B^{2}} & {\sigma_{I}^{2}/( {B\sqrt{2}} )} \\0 & 0 & 0 & {\sigma_{I}^{2}/( {B\sqrt{2}} )} & \sigma_{I}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 6} )\end{matrix}$

where σ_(D) is the uncertainty in the initial depth of the loweracceleration sensor 50 and σ_(I) is the uncertainty in the initialinclination of the lower acceleration sensor 50. The zero elements ofthe co-variance matrix Σ₀ result from the fact that the downhole portion30 is initially stationary (i.e., acceleration and velocity both equalzero).

The state vector X_(k-1) of epoch k−1 can be used to predict the statevector X_(k) of a later epoch k using the following equations:

a _(k) =a _(k-1);  (Eq. 7)

v _(k) =v _(k-1) +a _(k-1) *Δt;  (Eq. 8)

D _(k) =D _(k-1) +v _(k-1) *Δt+(a _(k-1) *Δt ²)/2;  (Eq. 9)

d _(k) =d _(k-1);  (Eq. 10)

I _(k) =I _(k-1) +d _(k-1) *v _(k-1) *Δt;  (Eq. 11)

where Δt is the time period between epoch k−1 and epoch k. In addition,certain other embodiments utilize other equations which includehigher-order terms to predict the state vector of epoch k based on anearlier state vector of epoch k−1.

The co-variance matrix χ for the predicted state vector is given by thefollowing diagonal matrix:

$\begin{matrix}{{\chi = \begin{bmatrix}( {p_{a}/\alpha} )^{2} & 0 & 0 & 0 & 0 \\0 & ( {p_{v}/\alpha} )^{2} & 0 & 0 & 0 \\0 & 0 & ( {p_{D}/\alpha} )^{2} & 0 & 0 \\0 & 0 & 0 & ( {p_{d}/\alpha} )^{2} & 0 \\0 & 0 & 0 & 0 & ( {p_{I}/\alpha} )^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 12} )\end{matrix}$

where p_(a) is the maximum change of acceleration over time period Δt,p_(v) is the maximum change of velocity over time period Δt, p_(D) isthe maximum change of depth over time period Δt, p_(d) is the maximumchange of apparent dogleg over time period Δt, and p₁ is the maximumchange of inclination over time period Δt. In certain embodiments, p_(a)is assumed to be given by p_(a)=2p_(D)(Δt)². In certain embodiments,p_(v) assumed to be given by p_(v)=p_(D)/Δt. In certain embodiments,p_(d) is assumed to be given by p_(d)=2p_(I)/B.

The parameter α provides a multiplication factor between the standarddeviation σ of a state vector element and the maximum change p of thestate vector element, such that the maximum change of the state vectorelement can be expressed as p=α*σ. In certain embodiments, themultiplication factor α is in a range between approximately 2 andapproximately 5, and in other embodiments, the multiplication factor αis substantially equal to 3.

The acceleration sensors provide the following measurements at epoch k:

A _(k) =[A _(L,k) A _(U,k)]^(T);  (Eq. 13)

where A_(L,k) is the measurement from the lower acceleration sensor 50and A_(U,k) is the measurement from the upper acceleration sensor 40 atepoch k. The co-variance matrix corresponding to the acceleration sensormeasurements at epoch k is provided by the following diagonal matrix:

$\begin{matrix}{{\Psi_{A,k} = \begin{bmatrix}\sigma_{A_{L},k}^{2} & 0 \\0 & \sigma_{A_{U},k}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 14} )\end{matrix}$

where σ_(A) _(L) _(,k) is the uncertainty of the lower accelerationsensor measurements and σ_(A) _(U) _(,k) is the uncertainty of the upperacceleration sensor measurements. In certain embodiments, σ_(A) _(I)_(,k) is the same for all epochs, and σ_(A) _(U) _(,k) is the same forall epochs. In certain embodiments in which the two acceleration sensorsare substantially identical, σ_(A) _(L) _(,k)=σ_(A) _(U) _(,k).

As discussed more fully below, in certain embodiments, additional aidingdata may be supplied. In certain embodiments, aiding velocitymeasurements [V_(k)] are provided with a corresponding co-variancematrix Ψ_(V,k)=[σ_(V,k) ²], where σ_(V,k) is the uncertainty of theaiding velocity measurements. In certain embodiments, aidingabsolute-depth measurements [S_(k)] are provided with a correspondingco-variance matrix Ψ_(S,k)=[σ_(S,k) ²], where σ_(S,k) is the uncertaintyof the aiding absolute-depth measurements. In certain embodiments,aiding relative-depth measurements [R_(k)] are provided with acorresponding co-variance matrix Ψ_(R,k)=[σ_(R,k) ²], where σ_(R,k) isthe uncertainty of the aiding relative-depth measurements. Other aidingmeasurements can be used in accordance with embodiments describedherein.

The theoretical acceleration sensor measurements can be calculated usingthe predicted state vector elements a_(k) and I_(k) in the followingequations:

A′ _(L,k) =a _(k) +g*cos(I _(k));  (Eq. 15)

A′ _(U,k) =a _(k) +g*cos(I _(k) −d _(k) *B);  (Eq. 16)

where A′_(L,k) is the theoretical lower acceleration sensor measurement,A′_(U,k) is the theoretical upper acceleration sensor measurement.

The equations which provide the predicted state vector at epoch k basedon the state vector at epoch k−1 can be expressed as the followingprediction matrix Φ_(k):

$\begin{matrix}{\Phi_{k} = {\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\{\Delta \; t} & 1 & 0 & 0 & 0 \\{( {\Delta \; t} )^{2}/2} & {\Delta \; t} & 1 & 0 & 0 \\0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & {v_{k - 1}*\Delta \; t} & 1\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 17} )\end{matrix}$

In this way, the predicted state vector at epoch k can be expressed asX_(k)=Φ_(k)*X_(k-1).

The prediction matrix Φ_(k) can not be used for a prediction of thestate co-variance matrix because it is non-linear (i.e., one of thestate elements is included in the matrix). Instead, a linear predictionmatrix Γ_(k) can be used to update the state co-variance matrix,corresponding to the uncertainty of the new state vector, as follows:

Σ_(k)=Γ_(k)*Σ_(k-1)*Γ_(k) ^(T)+χ;  (Eq. 18)

where Γ_(k) is given by the following:

$\begin{matrix}{\Gamma_{k} = {\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\{\Delta \; t} & 1 & 0 & 0 & 0 \\{( {\Delta \; t} )^{2}/2} & {\Delta \; t} & 1 & 0 & 0 \\0 & 0 & 0 & 1 & 0 \\0 & {d_{k - 1}*\Delta \; t} & 0 & {v_{k - 1}*\Delta \; t} & 1\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 19} )\end{matrix}$

Three matrices can be defined to be used to calculate updates of thestate vector X_(k) and the state co-variance matrix Σ_(k) based onmeasurements. The design matrix α_(A,k) corresponds to the partialderivatives of the theoretical measurements and is given by thefollowing:

$\begin{matrix}{\alpha_{A,k} = {\begin{bmatrix}1 & 0 & 0 & 0 & {{- g}*\sin \; ( I_{k -} )} \\1 & 0 & 0 & {{\sin ( {I_{k -} - {d_{k -}*B}} )}*B} & {- {\sin ( {I_{k -} - {d_{k -}*B}} )}}\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 20} )\end{matrix}$

The constant vector β_(A,k) corresponds to the theoretical measurementsminus the actual measurements and is given by the following:

$\begin{matrix}{\beta_{A,k} = {\begin{bmatrix}{A_{L,{k -}}^{\prime} - A_{L,k}} \\{A_{U,{k -}}^{\prime} - A_{U,k}}\end{bmatrix} = {\begin{bmatrix}{a_{k -} + {g*{\cos ( I_{k -} )}} - A_{L,k}} \\{a_{k -} + {g*{\cos ( {I_{k -} - {d_{k -}*B}} )}} - A_{U,k}}\end{bmatrix}.}}} & ( {{Eq}.\mspace{14mu} 21} )\end{matrix}$

The gain matrix G_(k) is given by the following:

G _(k)=Σ_(k-) *a _(A,k) ^(T)*(Ψ_(A,k)+α_(A,k)*Σ_(k-)*α_(A,k)^(T))⁻¹.  (Eq. 22)

The after-measurement update of the state vector X_(k) and theafter-measurement update of the state co-variance matrix Σ_(k) arecalculated as follows:

X _(k) =X _(k-) −G _(k)*β_(A,k);  (Eq. 23)

Σ_(k)=Σ_(k-) −G _(k)*α_(A,k)*Σ_(k-).  (Eq. 24)

where k− denotes the values of epoch k prior to the current update.Using such a labeling scheme, k− in Equations 20 through 24 denotes thevalues of the predicted state vector prior to the update using theacceleration measurements. Thus, X_(k-) and Σ_(k-) are the state vectorand state co-variance matrix, respectively, of epoch k after theprediction but prior to the measurement update.

Aiding Data

Stationary Data

In certain embodiments, the downhole portion 30 of the survey tool 10 isstopped within the wellbore 20 one or more times to obtain aiding datawhile the downhole portion 30 is stationary. In certain embodiments, thedownhole portion 30 is stopped at one or more random or arbitrary times,while in other embodiments, the downhole portion 30 is stopped atmultiple times with a predetermined period. In certain such embodimentswhich only have the first acceleration sensor 40 and the secondacceleration sensor 50, the state vector X_(k) and the co-variancematrix Σ_(k) can be expressed as:

$\begin{matrix}{{X_{k} = \begin{bmatrix}0 & 0 & D_{L,{k -}} & d_{k -} & I_{L,{k -}}\end{bmatrix}^{T}};} & ( {{Eq}.\mspace{14mu} 25} ) \\{{\Sigma_{k} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \Sigma_{33,{k -}} & \Sigma_{34,{k -}} & \Sigma_{35,{k -}} \\0 & 0 & \Sigma_{43,{k -}} & \Sigma_{44,{k -}} & \Sigma_{45,{k -}} \\0 & 0 & \Sigma_{53,{k -}} & \Sigma_{54,{k -}} & \Sigma_{55,{k -}}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 26} )\end{matrix}$

which in certain embodiments is given by:

$\begin{matrix}{{\Sigma_{k} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \sigma_{D,{k -}}^{2} & \sigma_{{Dd},{k -}} & \alpha_{{DI},{k -}} \\0 & 0 & \sigma_{{dD},{k -}} & \sigma_{d,{k -}}^{2} & \sigma_{{dI},{k -}} \\0 & 0 & \sigma_{{ID},{k -}} & \sigma_{{Id},{k -}} & \sigma_{I,{k -}}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 27} )\end{matrix}$

where k− in Equations 25 through 27 denotes the values of the variouselements prior to the update using the aiding stationary data. In othersuch embodiments which have cross-axial acceleration sensors, the statevector X_(k) and the co-variance matrix Σ_(k) can be expressed using thefollowing:

$\begin{matrix}{{d_{k} = {( {{\arctan ( {{- {HA}_{L,k}}/A_{L,k}} )} - {\arctan ( {{- {HA}_{U,k}}/A_{U,k}} )}} )/B}};} & ( {{Eq}.\mspace{14mu} 28} ) \\{{I_{k} = {\arctan ( {{- {HA}_{L,k}}/A_{L,k}} )}};} & ( {{Eq}.\mspace{14mu} 29} ) \\{{\Sigma_{k} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \Sigma_{33,{k -}} & 0 & 0 \\0 & 0 & 0 & {\sigma_{I}^{2}/B^{2}} & {\sigma_{I}^{2}/( {B\sqrt{2}} )} \\0 & 0 & 0 & {\sigma_{I}^{2}/( {B\sqrt{2}} )} & \sigma_{I}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 30} )\end{matrix}$

which in certain embodiments is given by:

$\begin{matrix}{\Sigma_{k} = {\begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \sigma_{D}^{2} & 0 & 0 \\0 & 0 & 0 & {\sigma_{I}^{2}/B^{2}} & {\sigma_{I}^{2}/( {B\sqrt{2}} )} \\0 & 0 & 0 & {\sigma_{I}^{2}/( {B\sqrt{2}} )} & \sigma_{I}^{2}\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 31} )\end{matrix}$

Velocity Data

In certain embodiments, aiding velocity data is provided by one or moresupplementary sensors which are part of the downhole portion 30 of thesurvey tool 10. As described above in relation to FIG. 3, examplesupplementary sensors include, but are not limited to, gamma-ray sensorsadapted to detect gamma rays from geological formations in proximity tothe downhole portion 30 and magnetic sensors adapted to detect casingcollars in proximity to the downhole portion 30. Such supplementarysensors do not continuously measure the velocity of the survey tool 10,but they detect landmark features which can be used to calculate theaverage velocity of the downhole portion 30 as the downhole portion 30passes the landmark feature or between the locations of two or moreselected landmark features.

FIG. 5A schematically illustrates a tool 10 having a first gamma-raysensor 80 and a second gamma-ray sensor 70 with the downhole portion 30moving past a geological formation 90 having a gamma-ray emission higherthan the surrounding formations. FIG. 5B schematically illustrates thesignals from the first gamma-ray sensor 80 and the second gamma-raysensor 70 as functions of time. The first gamma-ray sensor 80 is lowerthan the second gamma-ray sensor 70, so the first gamma-ray sensor 80detects the formation 90 before the second gamma-ray sensor 70 detectsthe same formation 90. By correlating the measurements of the twosupplementary sensors, and knowing the distance B_(v) between the twosupplementary sensors, the average velocity of the downhole portion 30as it passed the geological formation 90 can be calculated as follows:

V _(k) =B _(v)/(T _(L) −T _(U));  (Eq. 32)

where β_(v) is the distance between the first gamma-ray sensor 80 andthe second gamma-ray sensor 70, T_(L), is the time that the firstgamma-ray sensor 80 detects the geological formation 90, and T_(U) isthe time that the second gamma-ray sensor 70 detects the geologicalformation 90. Similarly, two magnetic sensors of the downhole portion 30can be used to determine an average velocity of the downhole portion 30relative to a casing collar of the wellbore 20.

The measured velocity V_(k) is the average velocity over the timeinterval ranging from T_(L) to T_(U). However, the average velocityapproaches the instantaneous velocity as the time interval (T_(L)−T_(U))approaches zero. In certain embodiments, the average velocity is used toanalyze the state vector X_(k). In certain other embodiments, theanalysis of the state vector X_(k) utilizes the instantaneous velocityat epoch k. In certain such embodiments, the distance B_(v) is selectedto be as small as possible without letting the noise or uncertainty inthe two time measurements corrupt the measured velocity.

The uncertainty σ_(V,k) in the measured velocity at epoch k depends onthe instability of the velocity over the time interval between T_(L),and T_(U), and on the uncertainty in the repeated detection of thelandmark feature being referenced. An estimate of the uncertainty of themeasured velocity is given by:

σ_(V,k) ²=(p _(v)*α*(T _(U) −T _(L))/Δt)²+2σ_(det,k) ²/(T _(U) −T_(L))²;  (Eq. 33)

where σ_(det,k) is the uncertainty in the detection of the actuallocation of the formation at epoch k, and p_(v), is the maximum changeof the velocity of the time period Δt of the epoch k. In certainembodiments, σ_(det,k) is constant among different epochs. In certainembodiments, the maximum change of the velocity over the time period Δtis assumed to be equal to the maximum change of the depth p_(D) dividedby the time period Δt, where Δt is small.

In certain embodiments in which such aiding velocity data is available,a design velocity vector α_(V,k) (corresponding to the partialderivatives of the theoretical velocity measurements which are equal tothe current state velocity) and a constant velocity vector β_(V,k)(corresponding to the theoretical measurements minus the actualmeasurements) can be expressed as:

α_(V,k)[0 1 0 0 0];  (Eq. 34)

β_(V,k) =[v _(k) −V _(k)].  (Eq. 35)

Using these two vectors, a gain velocity matrix G_(k) can be expressedas:

G _(k)=Σ_(k-)*α_(V,k) ^(T)*(Ψ_(V,k)+α_(V,k)*Σ_(k-)*α_(V,k)^(T))⁻¹;  (Eq. 36)

which can be used to express the state vector X_(k) and the co-variancematrix Σ_(k) as:

X _(k) =X _(k-) −G _(k)*β_(V,k);  (Eq. 37)

Σ_(k)=Σ_(k-) −G _(k)*α_(V,k)*Σ_(k-);  (Eq. 38)

where k− in Equations 35 through 38 denotes the values of the variouselements prior to the update using the aiding velocity data.

Absolute Depth

In certain embodiments in which the absolute depth of a landmark featureis known, the detection of this landmark feature by one or more sensorsof the downhole portion 30 can be used to provide aiding data. Incertain such embodiments, the time at which the landmark feature ispassed by the downhole portion 30 is noted, and for the correspondingepoch k, the depth S_(k) can be expressed as:

S _(k) =S _(k) ^(S) +B ^(S);  (Eq. 39)

where S_(k) ^(S) is the depth of the landmark feature and B^(S) is thedistance between the lower acceleration sensor 50 and the supplementarysensor in proximity to the lower acceleration sensor 50 (e.g., thesecond gamma-ray sensor 80 as in FIG. 3).

In certain embodiments, one or more of the same supplementary sensorsare used to provide both aiding velocity data and aiding absolute-depthdata. In certain embodiments, two supplementary sensors are used toprovide two separate absolute-depth measurements of the same landmarkfeature. In certain such embodiments, the two measured absolute depths,S_(k) and S_(k′), are related to two different epochs which aretemporally close, and are related by the following:

S _(k) =S _(k) ^(S) +B ^(SL);  (Eq. 40)

S _(k′) =S _(k) ^(S) +B ^(SU);  (Eq. 41)

where B^(SL) is the distance between the lower acceleration sensor 50and the second supplementary sensor (e.g., the second gamma-ray sensor80 as in FIG. 3), and B^(SU) is the distance between the loweracceleration sensor 50 and the first supplementary sensor (e.g., thefirst gamma-ray sensor 70 as in FIG. 3).

The uncertainty σ_(S,k) of the absolute-depth measurements at epochs kand k′ depends on both the uncertainty in the given value of theabsolute depth of the landmark feature and the uncertainty in thedetection of the landmark feature by the supplementary sensor. Theuncertainty of the absolute-depth measurement can be expressed asfollows:

σ_(S,k) ²=σ_(det,k) ²+σ_(given,k) ²;  (Eq. 42)

where σ_(det,k) is the uncertainty in the detection of the landmarkfeature and σ_(given,k) is the uncertainty in the given value of theabsolute depth of the landmark feature. In certain embodiments,a_(det,k) is constant for different epochs.

In certain embodiments in which aiding absolute-depth data is available,a design absolute-depth vector α_(S,k), a constant absolute-depth vectorβ_(S,k), and a gain absolute-depth matrix G_(k) can be used to expressthe state vector X_(k) and the co-variance matrix Σ_(k) as follows:

α_(S,k)=[0 0 1 0 0];  (Eq. 43)

τ_(S,k) =[D _(k-) −S _(k)];  (Eq. 44)

G _(k)=Σ_(k-)*α_(S,k) ^(T)*(Ψ_(S,k)+α_(S,k)*Σ_(k-)*α_(S,k)^(T))⁻¹;  (Eq. 45)

X _(k) =X _(k-) −G _(k)*β_(S,k);  (Eq. 46)=

Σ_(k)=ρ_(k-) −G _(k)*α_(S,k)*Σ_(k-);  (Eq. 47)

where k− in Equations 44 through 47 denotes the values of the variouselements prior to the update using the aiding absolute-depth data.

Relative Depth

In certain embodiments, the downhole portion 30 provides relative-depthdata which does not depend on the known depths of landmark features. Incertain embodiments, such relative-depth data is provided by twosupplementary sensors adapted to provide large and sharp signal spikesat landmark locations. In certain embodiments, these landmark locationsare previously known, while in other embodiments, these landmarklocations are previously unknown. In certain embodiments, therelative-depth data is provided by a similar configuration of twosupplementary sensors of the downhole portion 30 as that which providesthe aiding velocity data described above. However, while the velocitydata is advantageously provided by two supplementary sensors which arerelatively close together, the relative-depth data is advantageouslyprovided by two supplementary sensors which are farther apart from oneanother.

In certain embodiments, the epoch k is defined to be the epoch duringwhich the second supplementary sensor (e.g., the second gamma-ray sensor80 of FIG. 3) passes a detectable landmark location and the epoch k′ isdefined to be the epoch during which the first supplementary sensor(e.g., the first gamma-ray sensor 70 of FIG. 3) passes the samedetectable landmark location. The aiding relative-depth measurementR_(k) at epoch k of certain embodiments can be expressed as:

R _(k) =D _(k′) +B ^(r) +B ^(a);  (Eq. 48)

where D_(k′) is the calculated depth of the downhole portion 30 at epochk′, B^(r) is the distance between the first supplementary sensor and thesecond supplementary sensor (e.g., the two gamma-ray sensors 70, 80 ofFIG. 3), and B^(a) is the distance between the second accelerationsensor 50 and the second supplementary sensor (e.g., the secondgamma-ray sensor 80 of FIG. 3).

In certain embodiments, the uncertainty in the relative-depthmeasurement at epoch k depends on both the uncertainty of the calculateddepth at epoch k′ and the uncertainty in the relative depth detection.The uncertainty in the relative-depth measurement can be expressed asfollows:

σ_(R,k) ²=σ_(rel) ²+σ_(last,k);  (Eq. 49)

where σ_(rel) is the uncertainty in the relative-depth detection for thelandmark location and σ_(last,k) is the uncertainty of the calculateddepth at epoch k. In certain embodiments, σ_(last,k) is given by thethird diagonal element (Σ_(33,k′)) of the solution co-variance matrix atepoch k′. In certain embodiments, σ_(last,k) is equal to σ_(D), which isthe uncertainty in the depth measurement.

In certain embodiments in which aiding relative depth data is available,a design relative-depth vector α_(R,k), a constant relative-depth vectorβ_(R,k), and a gain relative-depth matrix G_(k) can be used to expressthe state vector X_(k) and the co-variance matrix Σ_(k) as follows:

α_(R,k)=[0 0 1 0 0];  (Eq. 50)

β_(R,k) =[D _(k-) −R _(k)].  (Eq. 51)

G _(k)=Σ_(k-)*α_(R,k) ^(T)*(Ψ_(R,k)+α_(R,k)*Σ_(k-)*α_(R,k)^(T))⁻¹;  (Eq. 52)

X _(k) =X _(k-) −G _(k)*β_(R,k);  (Eq. 53)

Σ_(k)=Σ_(k-) −G _(k)*α_(R,k)*Σ_(k-);  (Eq. 54)

where k− in Equations 51 through 54 denotes the values of the variouselements prior to the update using the aiding relative-depth data.

Bend Data

Certain embodiments described above may experience limited systemperformance. For example, in an open wellbore, the passage of the toolpast a formation feature exhibiting a significant formation radiationvariation can be detected by a pair of gamma-ray detectors spaced aknown distance apart along the tool string. However, in certainembodiments, the radiation signal variation can be small, e.g., whenpassing through a large uniform section of formation, or when drillingparallel to the formation strata (as is done in the construction of anextended reach wellbore. Under such conditions, the performance of thedepth system in certain embodiments may degrade substantially and theaccuracy of the depth estimates generated may be poor. In addition,while the passage of the tool past a casing joint can provide awell-defined magnetic disturbance that can be detected by a pair of CCLsspaced a known distance apart along the tool string, such CCLs are oflimited use in open wellbores not having the casing joints. Certainembodiments described below advantageously overcome such limitations.

FIG. 6 schematically illustrates an example survey tool 10 in accordancewith certain embodiments described herein. The survey tool 10 has a bendsensor 100 which generates a third signal indicative of an amount ofbend of at least a portion of the downhole portion 30. In certainembodiments, the bend sensor 100 is between the first supplementarysensor 40 and the second supplementary sensor 50. As shown in FIG. 6,the bend sensor 100 can be located approximately midway or equidistantbetween the first acceleration sensor 40 and the second accelerationsensor 50. The amount of bend or well curvature in certain embodimentscan be used to enhance the measurement accuracy of estimations ofacceleration, velocity, and along-hole position or depth as the downholeportion 30 traverses the path of the wellbore 20.

The bend sensor 100 of certain embodiments can determine the curvatureof the downhole portion 30 using various methods by measuring a physicalcharacteristic of the downhole portion 30 that changes as the downholeportion 30 bends. For example, the bend sensor 100 of certainembodiments is sensitive to mechanical strain in a part of the downholeportion 30 produced upon bending a part of the downhole portion 30(e.g., a tubular casing of the downhole portion 30). In certainembodiments, the bend sensor 100 is sensitive to deformation,deflection, or movement of at least a part of the downhole portion 30relative to another part of the downhole portion 30. In certainembodiments, the bend sensor 100 comprises one or more piezoelectric orpiezoresistive elements which are mounted in the downhole portion 30 andwhich expand or contract in response to bending of the downhole portion30. In certain other embodiments, the bend sensor 100 providesultrasonic measurements of the tubular casing as it bends. In certainother embodiments, the bend sensor 100 comprises a foil-type sensorcomprising a metallic foil pattern supported by an insulated flexiblebacking. Deformation of the foil causes its electrical resistance tochange, which can be measured (e.g., by a Wheatstone bridge) and themeasured resistance is related to the mechanical strain which causes thedeformation. Other types of bend sensors 100 are also compatible withvarious embodiments described herein.

In certain embodiments, the bend sensor 100 includes an optical system110 responsive to bending of the downhole portion 30, as schematicallyillustrated by FIGS. 7A and 7B. The optical system 110 of certainembodiments includes a light source 112 and a light detector 114separated from the light source 112 by a non-zero distance L_(d) alongthe wellbore 20. In certain embodiments, the light source 112 compriseseither a light-emitting diode or a semiconductor laser diode which emitsa narrow light beam 116 that impinges on the light detector 114. Incertain embodiments, the light detector 114 comprises a photosensorwhich generates a signal indicative of the position at which the lightbeam 116 impinges the on light detector 114. Examples of photodetectorscompatible with certain embodiments described herein include, but arenot limited to, arrays of photoresistors, photovoltaic cells,photodiodes, and charge-coupled devices (CCDs). The optical componentsare advantageously both mechanically rugged and capable of operating atthe high temperatures that can be expected downhole during drilling andwell survey operations (e.g., 150° C. or more). In certain embodiments,the non-zero distance L_(d) between the light source 112 and the lightdetector 114 is greater than about 2 meters, while in certain otherembodiments, the non-zero distance L_(d) is in a range between about 1meter and about 3 meters.

As illustrated schematically by FIG. 7A, when the optical system 110 isin an unbent state (e.g., when the downhole portion 30 is in arelatively straight section of the wellbore 20), the light 116 impingesupon a first portion 117 of the light detector 114. In certainembodiments, the first portion 117 is approximately at the center of thelight detector 114. As schematically illustrated by FIG. 7B, when theoptical system 110 is in a bent state (e.g., when the downhole portion30 is in a curved portion of the wellbore 20) however, the light 116impinges on a second portion 118 of the light detector 104. In certainembodiments, the second portion 118 is displaced from the center of thelight detector 114. The displacement 119 between the first portion 117and the second portion 118 is dependent on the amount of bend of thebend sensor 100 (e.g., between the portion of the downhole portion 30containing the light source 112 and the portion of the downhole portion30 containing the light detector 114. In certain embodiments, the bendsensor 100 generates the third signal in response to the distance 119between the first portion 117 and the second portion 118. In certainother embodiments, the bend sensor 100 generates the third signal inresponse to the position of the second portion 118.

The laser beam 116 of certain embodiments is directed generally parallelto the downhole portion 30 when the downhole portion 30 is in arelatively straight section of the wellbore 20, as shown schematicallyby FIG. 7A. In certain such embodiments, the laser beam 116 is generallycollinear with a longitudinal axis of the downhole portion 30. Incertain other embodiments, the laser beam 116 is directed generallynon-parallel to the downhole portion 30 when the downhole portion 30 isin a relatively straight section of the wellbore 20.

In certain embodiments in which the downhole portion 30 bends uniformlywith the curvature of the wellbore 20, the displacement 119 between thesecond portion 118 and the first portion 117 is proportional to the bendof the downhole portion 30, and hence provides a direct measure of thewell dogleg curvature, D. For small angles of curvature, the doglegcurvature D can be expressed in terms of the displacement 119(represented by x) and the separation, L_(d), between the light source112 and the light detector 114 as D≈2x/L_(d). Thus, in certainembodiments, the signal generated by the light detector 114 can be usedwith the known distance between the light source 112 and the lightdetector 114 to calculate the well dogleg curvature D.

While FIGS. 7A and 7B schematically illustrate an example optical system110 compatible with certain embodiments described herein, other opticalsystems are also compatible with certain embodiments described herein.For example, a commercial system, known as MAXIBOR® marketed by ReflexInstrument AB, may be adapted to provide the required measurementinformation in accordance with certain embodiments described herein.Other example optical systems 110 using interferometry (e.g., asdescribed by U.S. Pat. Nos. 6,023,325 and 5,946,094, both of which areincorporated in their entireties by reference herein) can be used tomeasure curvature in accordance with certain embodiments describedherein.

In certain embodiments, the third signal generated by the bend sensor100 is indicative of an amount of bend in each of two generallyorthogonal vertical planes. The components of the displacement 119 ineach of these two planes can be used to determine the amount of bend ineach of these planes. For example, FIG. 7B schematically shows the bendsensor 100 bent along a first direction in a first vertical plane andhaving a displacement 119 (represented by x) along the light detector114 in the first plane. Similarly, the bend of the bend sensor 100 in asecond vertical plane generally orthogonal to the first vertical planecan be measured by detecting a displacement y along a second directiongenerally orthogonal to the first direction.

In certain embodiments, the accelerometer measurements made by the firstand second acceleration sensors 40, 50 while the downhole portion 30 isat two positions can be used to provide an estimate of the bend of thedownhole portion 30 in a vertical plane defined by the two positions. Incontrast, the bend sensor 100 according to certain embodiments describedherein advantageously measures the total curvature of the downholeportion 30, regardless of the vertical plane defined by the positions atwhich accelerometer measurements are made. The estimates of the bend inthe vertical plane of the two accelerometer measurement positions and inthe total curvature can be compared to one another in certainembodiments as part of a wellbore drilling system control algorithm.While any bending out of the vertical plane defined by the twoaccelerometer measurement positions is typically small as compared tobending in this vertical plane, certain embodiments advantageously allowthe components of the bend in this plane and orthogonal to this plane tobe calculated and compared as part of the system control algorithm. Incertain embodiments, the device is configured to separate the componentsof the bend in two directions with respect to the principal axes of thetool 10. Based on the accelerometer measurements and their mountingorientation with respect to the principal axes of the tool 10, thevertical and horizontal components of the bend may then be calculated.The vertical component measurement or both of the two bend componentmeasurements can then be used in certain embodiments as input to thewellbore drilling system control algorithm to direct the drilling systemin a desired direction.

In certain embodiments, the survey tool 10 has a controller 60 tocalculate a depth, a velocity, or both a depth and a velocity of thedownhole portion 30 in response to the first, second, and third signals.In certain embodiments, the measurements are taken simultaneously by thefirst acceleration sensor 40, the second acceleration sensor 50, and thebend sensor 100. In certain other embodiments, the controller acceptssimultaneous measurements from the first and second acceleration sensors40, 50 and the bend sensor 100, all of which are taken at regularintervals of time. Both accelerometer and dogleg measurements can beutilized at the same instants of time, which simplifies the controlalgorithm and can enhance its effectiveness by reducing the effect ofaccelerometer correlations at the main updates. In certain embodiments,the combination of dual accelerometer measurements with aiding data fromthe bend sensor 100 allows accurate estimates of the depth and velocityof the survey tool 10 to be generated as it travels along the wellbore.

FIG. 8 shows a flowchart of an example method 300 of determining a depthor a velocity, or both a depth and a velocity of a downhole portion 30of a tool 10 movable within a wellbore 20 in accordance with certainembodiments described herein. The method 300 comprises providing a tool10 in an operational block 310. The tool 10 comprises a downhole portion30 movable within the wellbore 20 in a direction generally parallel tothe wellbore 20. The tool 10 further comprises a first accelerationsensor 40 mounted at a first position within the downhole portion 30.The first acceleration sensor 40 generates a first signal indicative ofa first acceleration in a first direction generally parallel to thewellbore 20 at the first position. The tool 10 further comprises asecond acceleration sensor 50 mounted at a second position within thedownhole portion 30. The second acceleration sensor 50 generates asecond signal indicative of a second acceleration in a second directiongenerally parallel to the wellbore 20 at the second position. The tool10 further comprises a bend sensor 100 generating a third signalindicative of an amount of bend of at least a portion of the downholeportion 30.

The method 300 further comprises generating the first signal, the secondsignal, and the third signal while the downhole portion 30 is at thefirst location within the wellbore 20 in an operational block 320. Themethod 300 further comprises generating the first signal, the secondsignal, and the third signal while the downhole portion 30 is at thesecond location within the wellbore 20 in an operational block 330. Themethod 300 further comprises calculating a depth, or a velocity, or botha depth and a velocity, of the downhole portion 30 of the tool 10 inresponse to the first, second, and third signals generated while thedownhole portion 30 is at the first location and the first, second, andthird signals generated while the downhole portion 30 is at the secondlocation.

FIG. 9 is a flowchart of an example method 400 of determining a depth ora velocity, or both a depth and a velocity of a downhole portion 30 of atool 10 movable within a wellbore 200 in accordance with certainembodiments described herein. The method 400 comprises receiving one ormore acceleration measurements from at least one acceleration sensor inthe downhole portion 30 of the tool 10 in an operational block 410. Themethod 400 further comprises receiving one or more measurements of anamount of bend of at least a portion of the downhole portion 30 in anoperational block 420. The method 400 further comprises calculating adepth, or a velocity, or both a depth and a velocity of the downholeportion 30 of the tool 10 in response to the one or more accelerationmeasurements and the one or more measurements of the amount of bend.

As with the example embodiment described above, another exampleembodiment is described below in which the periodicity of themeasurements from the two accelerometers define time periods or “epochs”whereby one set of accelerometer measurements are taken at every epochk. Aiding data from the bend sensor 100 is advantageously included incertain embodiments at a subset of these epochs to enhance the resultantaccuracy of the results. For example, the dogleg at epoch k, d_(k), whenepoch k is an acceleration/dogleg update epoch, may be determined from asingle dogleg measurement or by averaging a number of doglegmeasurements obtained at discrete intervals between the last and currentmeasurement update time. The uncertainty in the dogleg measurement atepoch k, σ_(d,k), is dependent on the instability of the bend sensor 100over the time interval of the epoch k (e.g., from t_(l) to t_(u)).

Another Example Embodiment Utilizing Dogleg Data

The example embodiment described below calculates the depth, thevelocity, or both the depth and the velocity of the downhole portion 30using a recursive estimation algorithm employing measurements obtainedat multiple locations to update estimates of the depth, the velocity, orboth the depth and the velocity of the downhole portion 30. The statevector X_(k) and the state co-variance matrix at epoch k can beexpressed by Equations 1 and 2, respectively. The co-variance matrix Σ₀for the initial state at epoch k=0 can be expressed as the following:

$\begin{matrix}{{\Sigma_{0} = \begin{bmatrix}0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & \sigma_{D}^{2} & 0 & 0 \\0 & 0 & 0 & \sigma_{d}^{2} & 0 \\0 & 0 & 0 & 0 & \sigma_{I}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 55} )\end{matrix}$

where σ_(D) is the uncertainty in the initial depth of the loweracceleration sensor 50, σ_(d) is the uncertainty in the initial dogleg,and σ_(I) is the uncertainty in the initial inclination of the loweracceleration sensor 50. The zero elements of the co-variance matrix Σ₀result from the fact that the downhole portion 30 is initiallystationary (i.e., acceleration and velocity both equal zero).

Equations 7 through 11 can be used to predict the state vector X_(k) ofa later epoch k from the state vector X_(k-1) of epoch k−1. Theco-variance matrix χ for the predicted state vector is given by thediagonal matrix of Equation 12.

The acceleration sensors 40, 50 and the bend sensor 100 provide thefollowing measurements at epoch k:

A _(k) =[A _(L,k) A _(U,k) d _(k)]^(T);  (Eq. 56)

where A_(L,k) is the measurement from the lower acceleration sensor 50,A_(U,k) is the measurement from the upper acceleration sensor 40 atepoch k, and d_(k) is the measurement from the bend sensor 100.

The co-variance matrix corresponding to the measurements at epoch k isprovided by the following diagonal matrix:

$\begin{matrix}{{\Psi_{A,k} = \begin{bmatrix}\sigma_{A_{L},k}^{2} & 0 & 0 \\0 & \sigma_{A_{U},k}^{2} & 0 \\0 & 0 & \sigma_{d}^{2}\end{bmatrix}};} & ( {{Eq}.\mspace{14mu} 57} )\end{matrix}$

where σ_(A) _(L) _(,k) is the uncertainty of the lower accelerationsensor measurements, σ_(A) _(U) _(,k) is the uncertainty of the upperacceleration sensor measurements, and σ_(d) is the uncertainty of thebend sensor measurements. In certain embodiments, σ_(A) _(L) _(,k) isthe same for all epochs, σ_(A) _(U) _(,k) is the same for all epochs,and σ_(d) is the same for all epochs. In certain embodiments in whichthe two acceleration sensors are substantially identical, σ_(A) _(L)_(,k)=σ_(A) _(U) _(,k).

The theoretical acceleration sensor measurements can be calculated usingthe predicted state vector elements a_(k) and I_(k) in the followingequations:

A′ _(L,k) =a _(k) +g*cos(I _(k-1));  (Eq. 58)

A′ _(U,k) =a _(k) +g*cos(I _(k-1) −d _(k-1) *B);  (Eq. 59)

d′ _(k) =d′ _(k-1);  (Eq. 60)

where A′_(L,k) is the theoretical lower acceleration sensor measurement,A′_(U,k) is the theoretical upper acceleration sensor measurement, andd′_(k-1) is the theoretical dogleg measurement. The predicted statevector at epoch k can be expressed as X_(k)=Φ_(k)*X_(k-1), using theprediction matrix of Equation 17, and the linear prediction matrix ofEquation 19 can be used in Equation 18 to update the state co-variancematrix.

The design matrix β_(A,k) corresponds to the partial derivatives of thetheoretical measurements and is given by the following:

$\begin{matrix}{\alpha_{A,k} = {\begin{bmatrix}1 & 0 & 0 & 0 & {{- g}*{\sin ( I_{k} )}} \\1 & 0 & 0 & {g*{\sin ( {I_{k} - {d_{k}*B}} )}*B} & {{- g}*{\sin ( {I_{k} - {d_{k}*B}} )}} \\0 & 0 & 0 & 1 & 0\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 61} )\end{matrix}$

The constant vector β_(A,k) corresponds to the theoretical measurementsminus the actual measurements and is given by the following:

$\begin{matrix}{\beta_{A,k} = {\begin{bmatrix}{a_{k} + {g*{\cos ( I_{k} )}} - A_{L,k}} \\{a_{k} + {g*{\cos ( {I_{k} - {d_{k}*B}} )}} - A_{U,k}} \\{d_{k}^{\prime} - d_{k}}\end{bmatrix}.}} & ( {{Eq}.\mspace{14mu} 62} )\end{matrix}$

The gain matrix G_(k) is given by the following:

G _(k)=Σ_(k)*α_(A,k) ^(T)*(Ψ_(A,k)+α_(A,k)*Σ_(k)*α_(A,k) ^(T))⁻¹.  (Eq.63)

The after-measurement update of the state vector X_(k) and theafter-measurement update of the state co-variance matrix Σ_(k) arecalculated as shown in Equations 23 and 24.

Other types and/or combinations of aiding data can also be used inaddition to the bend or dogleg data in certain other embodiments. Incertain embodiments, the aiding data can include the rate of change ofthe dogleg measurements. In certain embodiments, the aiding data caninclude the inclination determination based on summation of verticalcomponents of the measured dogleg increments.

Certain embodiments described herein advantageously address thelong-standing problem of inaccurate wellbore depth measurements byproviding accurate and reliable depth measurements in a wellbore for usein various aspects of oil exploration and production. Certainembodiments described herein advantageously provide the depth andvelocity of a package of geophysical and navigational instruments inreal time as the package is lowered and/or raised in a borehole, withoutmaking use of any surface measurements. Certain such embodiments providea measure of depth based entirely on the use of down-hole sensors, andare independent of any surface measurement devices, including wire-linedepth, wire-line velocity, or pipe tally depth measurements, each ofwhich is subject to error in the detection of true down-hole locationand movement.

In certain embodiments, acceleration is sensed along the length of thewellbore (e.g., in the z-direction) and is corrected for the gravitycomponent to yield an estimate of the acceleration with respect to thewellbore. This quantity can be integrated once to yield an estimate ofalong-hole velocity and integrated a second time to obtain an estimateof depth along the wellbore. In a curved section of the wellbore, thetwo accelerometers of certain embodiments nominally generate the samemeasurement when at the same location within the wellbore, thusproviding an indication of when the tool has travelled the knownseparation distance between the two accelerometers.

In certain embodiments, a processing algorithm based on a mathematicalmodel of the along-hole trajectory is used to provide estimates of theacceleration, velocity, depth, well curvature (dogleg), and inclinationof the wellbore. The measurements generated by the two accelerometers incertain embodiments can be compared with estimates of the samequantities derived from the states of the model. These measurementdifferences can form the inputs to the processing algorithm whicheffectively cause the outputs of the model to be driven into coincidencewith the measurements, thus correcting the outputs of the model.

In certain embodiments, the estimates of wellbore depth in doglegwellbore sections are substantially improved by utilizing additionalpairs of sensors (e.g., gamma-ray sensors, casing collar locators orCCLs, gyroscopes, and/or accelerometers) built into the tool string toaid the measurement process as the tool string moves along the wellboreby detecting small but often distinct mechanical disturbances. By usingpairs of sensors mounted a known distance apart along the tool string,certain embodiments can detect the same formation feature (and/or casingjoint in the case of CCLs). The known separation between the two sensorscan be used in certain embodiments to extract either a measured speed ofthe tool along the wellbore, or the incremental distance moved as thetwo sensors pass the same point in the well at different times. Incertain such embodiments, the measurements are affected by the timingresolution of the respective measurements by each sensor to accuratelyresolve the time elapsed between the second sensor detecting a featureand the earlier detection of the feature by the first sensor.

In certain embodiments, the estimates of acceleration, velocity, andalong-hole position or depth are supplemented by the measurements ofwell curvature (dogleg) and inclination at each measurement location asthe tool string traverses the path of the wellbore. The measurementaccuracy in certain such embodiments is enhanced by the use of theindependent measurements of well curvature or inclination, obtained inthe vicinity of the sensor locations, thereby increasing the accuracyand reliability of the estimation algorithm.

Various embodiments of the present invention have been described above.Although this invention has been described with reference to thesespecific embodiments, the descriptions are intended to be illustrativeof the invention and are not intended to be limiting. Variousmodifications and applications may occur to those skilled in the artwithout departing from the true spirit and scope of the invention asdefined in the appended claims.

What is claimed is:
 1. An apparatus comprising: a downhole portionconfigured to move along a wellbore, the downhole portion comprising: afirst acceleration sensor configured to generate at least one firstsignal indicative of a first acceleration of the first accelerationsensor; a second acceleration sensor spaced from the first accelerationsensor, the second acceleration sensor configured to generate at leastone second signal indicative of a second acceleration of the secondacceleration sensor; and a sensor configured to generate at least onethird signal indicative of a bend of the downhole portion; and acontroller configured to calculate a depth along the wellbore, avelocity along the wellbore, or both a depth and a velocity along thewellbore of the downhole portion in response to the at least one firstsignal, the at least one second signal, and the at least one thirdsignal and independently of wire-line or pipe-tally measurements.
 2. Theapparatus of claim 1, wherein the bend is between the first accelerationsensor and the second acceleration sensor.
 3. The apparatus of claim 2,wherein the first acceleration is not parallel to the secondacceleration when the bend is non-zero.
 4. The apparatus of claim 1,wherein the controller is configured to calculate the depth along thewellbore, the velocity along the wellbore, or both the depth and thevelocity along the wellbore of the downhole portion based entirely onthe use of down-hole sensors.
 5. The apparatus of claim 1, wherein thecontroller is configured to calculate the depth along the wellbore, thevelocity along the wellbore, or both the depth and the velocity alongthe wellbore of the downhole portion independently of errors fromsurface measurements.
 6. The apparatus of claim 1, wherein the bendcorresponds to a non-zero angle of the downhole portion between thefirst acceleration sensor and the second acceleration sensor.
 7. Theapparatus of claim 1, wherein the first and second acceleration sensorsare spaced apart from one another by a distance larger thanapproximately 10 meters.
 8. The apparatus of claim 1, wherein the sensoris mounted between the first acceleration sensor and the secondacceleration sensor.
 9. A method for generating information indicativeof a depth along a wellbore or a velocity along a wellbore or both adepth and a velocity along a wellbore of a tool configured to movewithin a wellbore, the method comprising: receiving at least one firstsignal indicative of a first acceleration of a first acceleration sensorof the tool; receiving at least one second signal indicative of a secondacceleration of a second acceleration sensor of the tool; receiving atleast one third signal indicative of a bend of the tool; storing atleast a portion of the at least one first signal, the at least onesecond signal, and the at least one third signal; and calculating adepth along the wellbore, or a velocity along the wellbore, or both adepth along the wellbore and a velocity along the wellbore of the toolin response to the at least one first signal, the at least one secondsignal, and the at least one third signal and independently of wire-lineor pipe-tally measurements.
 10. The method of claim 9, wherein the bendis between the first acceleration sensor and the second accelerationsensor.
 11. The method of claim 10, wherein the at least one firstsignal and the at least one second signal are indicative of the firstacceleration being not parallel to the second acceleration.
 12. Themethod of claim 9, wherein said calculating is not based on signals fromsensors that are outside of the wellbore.
 13. The method of claim 9,wherein said calculating is independent from errors from surfacemeasurements.
 14. The method of claim 9, wherein said calculatingcomprises using a recursive algorithm.
 15. The method of claim 14,wherein the recursive algorithm employs measurements obtained atmultiple locations to update estimates of the depth along the wellbore,the velocity along the wellbore, or both the depth along the wellboreand the velocity along the wellbore of the tool.
 16. A controllerconfigured to receive signals from a tool configured to move within awellbore, the controller comprising: one or more processors configuredto calculate a depth along a wellbore, or a velocity along the wellbore,or both a depth along the wellbore and a velocity along the wellbore ofthe tool, the one or more processors performing the calculation inresponse to at least one first signal, at least one second signal, andat least one third signal and independently of wire-line or pipe-tallymeasurements, wherein the at least one first signal is indicative of afirst acceleration of a first acceleration sensor of the tool, the atleast one second signal is indicative of a second acceleration of asecond acceleration sensor of the tool, and the at least one thirdsignal is indicative of a bend of the tool.
 17. The controller of claim16, wherein the bend is between the first acceleration sensor and thesecond acceleration sensor.
 18. The controller of claim 17, wherein theat least one first signal and the at least one second signal areindicative of the first acceleration being not parallel to the secondacceleration.
 19. The controller of claim 16, wherein at least a portionof the controller is located outside the wellbore and is operationallycoupled to the tool within the wellbore.
 20. The controller of claim 16,wherein the depth along the wellbore, the velocity along the wellbore,or both the depth and the velocity along the wellbore of the tool arenot based on signals from sensors that are outside the wellbore.
 21. Thecontroller of claim 20, wherein the controller is configured tocalculate the depth along the wellbore, the velocity along the wellbore,or both the depth and the velocity along the wellbore of the toolindependently of errors from surface measurements.